Ok, just want to post a heads-up since I wasted an hour of time figuring this out. I had logged onto my Alliance Toon and leveled the Moonkin (Alliance version). For some reason, Bliz has made two specific versions of the same pet and they cannot be used by the opposite faction. Once my teams were setup with the Battle Pet Teams Addon using the Alliance version of the pet, I was playing on my Horde toon for daily pet battles. I was unable to get any teams to swap or make any changes to my lineups.

Once I finally figured out my error, I re-logged onto my Alliance toon and removed the pet from the teams, went back to the Horde and the "issue" was fixed. Lesson learned, stick with the Horde:)